Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

semitagged division of ,

(1.A.3)

Let be any division of . If we take for , then the point-interval pair is a -fine tagged partial division of and, therefore, from (1.A.3) and since we can assume, without loss of generality (see comments in the paragraph before the statement), that for , we have

      and the proof is complete.

      Lemma 1.98: Suppose . The following properties are equivalent:

      1  is absolutely integrable;

      2 .

      Proof. (i) (ii). Suppose is absolutely integrable. Since the variation of , , is given by

      we have

(ii) (i). Suppose . We prove that the integral exists and . Given , we need to find a gauge on such that

      whenever is -fine. However,

(1.A.4)